There has been an increased interest in combining artificial neural networks and fuzzy systems. Fuzzy logic provides a mathematical foundation for approximate reasoning. Fuzzy logic controllers have proven very successful in a variety of applications such as control of roll and movement for a flexible wing aircraft, controller in a warm water plant, and traffic control. The parameters of adaptive fuzzy systems have clear physical meanings which facilitates the choice of their initial values. Furthermore, rule-based information can be incorporated into fuzzy systems in a systematic way.
Artificial neural networks emulate on a small scale the information processing mechanisms found in biological systems which are based on the cooperation of neurons which perform simple operations and on their ability to learn from examples. Artificial neural networks have become valuable computational tools in their own right for tasks such as pattern recognition, control, and forecasting.
Fuzzy systems and multilayer perceptions are computationally equivalent, i.e. they are both universal approximators. Recurrent neural networks have been shown to be computationally equivalent with Turing machines; whether or not recurrent fuzzy systems are also Turing equivalent remains an open question. While the methodologies underlying fuzzy systems and neural networks are quite different, their functional forms are often similar. The development of powerful learning algorithms for neural networks has been beneficial to the field of fuzzy systems which adopted some learning algorithms; e.g. there exist backpropagation training algorithms for fuzzy logic systems which are similar to the training algorithms for neural networks.
In some cases, neural networks can be structured based on the principles of fuzzy logic as described in the article by P. Goode et al entitled "A hybrid fuzzy/neural systems used to extract heuristic knowledge from a fault detection problem," in Proc. of the Third IEEE Conference on Fuzzy Systems, vol. III pp. 1731-1736, 1994 and in an article by C. Perneel et al entitled "Fuzzy Reasoning and Neural Networks for Decision Making Problems in Uncertain Environments" in Proc. of the Third IEEE Conference on Fuzzy Systems, vol. II, pp. 1111-1125, 1994. Neural network representations of fuzzy logic interpolation have also been used within the context of reinforcement learning.
A large class of problems where the current state depends on both the current input and the previous state can be modeled by finite-state automata or their equivalent grammars. The next step is to determine whether recurrent neural networks can also represent fuzzy finite-state automata (FFAs) and thus be used to implement recognizers of fuzzy regular grammars.
Fuzzy grammars have been found to be useful in a variety of applications such as in the analysis of X-rays, in digital circuit design, and in the design of intelligent human-computer interfaces. The fundamentals of FFAs have been discussed in articles by B. Gaines et al entitled "The Logic of Automata" in Int'l Journal of General Systems, vol. 2, pp. 191-208, 1976, by E. Santos entitled "Maximum Automata" in Information and Control, vol. 13, pp. 363-377, 1968 and by W. Wee et al entitled "A Formulation of Fuzzy Automata and its Applications as a Model of Learning Systems," in IEEE Transactions on System Science and Cybernetics, vol. 5, pp. 215-223, 1969, each without presenting a systematic method for machine synthesis. Neural network implementations of fuzzy automata have been proposed in an article by J. Grantner et al entitled "Synthesis and Analysis of Fuzzy Logic Finite State Machine Models," in Proc. of Third IEEE Conference on Fuzzy Systems, vol. I, pp. 205-210, 1994, and in another article by J. Grantner et al entitled "VLSI Implementations of Fuzzy Logic Finite State Machines," in Proc. of the Fifth IFSA Congress, pp. 781-784, 1993, and in an article by S. Lee et al entitled "Fuzzy Neural Networks," in Mathematical Biosciences, vol. 23, pp. 151-177, 1975, and an article by F. Unal et al entitled "A Fuzzy Finite State Machine Implementation Based on a Neural Fuzzy System," in Proc. of the Third Int'l Conf. on Fuzzy Systems, vol. 3, pp. 1749-1754, 1994. The synthesis method proposed by Grantner et al, supra, uses digital design technology to implement fuzzy representations of states and outputs. In Unal et al, supra, the implementation of a Moore machine with fuzzy inputs and states is realized by training a feedforward network explicitly on the state transition table using a modified backpropagation algorithm. The fuzzification of inputs and states reduces the memory size that is required to implement the automaton in a microcontroller, e.g. antilock braking systems. In related work, an algorithm for implementing weighted regular languages in neural networks with probabilistic logic nodes was discussed in an article by T. Ludermir entitled "Logical Networks Capable of Computing Weighted Regular Languages," in Proc. of the Int'l Joint Conf. on Neural Networks 1991, vol. 11, pp. 1687-1692, 1991. A general synthesis method for synchronous fuzzy sequential circuits has been discussed in an article by T. Watanabe et al entitled "Synthesis of Synchronous Fuzzy Sequential Circuits," in Proc. of the Third IFSA World Congress, pp. 288-291, 1989. A synthesis method for a class of discrete-time neural networks with multilevel threshold neurons with applications to gray level image processing has been proposed in an article by J. Si et al entitled "Analysis and Synthesis of a Class of Discrete-Time Neural Networks with Multilevel Threshold Neurons," in IEEE Trans. on Neural Networks, vol. 6, no. 1, p. 105, 1995.